The Chilean is into the last year of his contract and many expect him not to re-sign with the Gunners.
He told "The message from Wenger to Sanchez today was: 'Give me until January and then you can go'," Keown told the Daily Mail.
"Wenger is clearly anticipating that Sanchez will be very low when he returns from the international break.
"Sanchez has been trying to leave for the last year or so, was unable to get his move on deadline day and now Chile have not qualified for the World Cup.
"To finally hear that he can leave in a matter of months is the news that he will have been craving.
"Now, Wenger will be expecting to see a brightness in Sanchez's eye at training on Friday morning, not one of resignation.
"Sanchez is a lion who wants to play football and I do not expect him to sulk."